Calculate the approximate mass of the Milky Way Galaxy from the fact that the Sun orbits the galactic center every 230 million years at a distance of 27,000 light-years. (As dis-cussed in Chapter 19, this calculation tells us only the mass of the galaxy within the Sun’s orbit.)

The approximate mass of the Milky Way Galaxy within the Sun's orbit is .

Explanation:

The Universal law of gravitation shows the interaction of gravity between two bodies:

 (1)

Where G is the gravitational constant, M and m are the masses of the two objects and r is the distance between them.

For this particular case, M is the mass of the Milky Way Galaxy and m is the mass of the Sun. Since it is a circular motion the centripetal acceleration will be:

 (2)

Then Newton's second law () will be replaced in equation (1):

By replacing (2) in equation (1) it is gotten:

(3)

Therefore, the mass of the Milky Way Galaxy can be determined if M is isolated from equation (3):

(4)

But r is 27000 light years (r = 27000 ly), Notice that it is necessary to express r in units of meters:

⇒

Before using equation 4 it is necessary to find the velocity:

(5)

Where d is the distance and t is the time, for this case t = 230000000 years.

⇒

Then, the values of d and t can be replaced in equation 5:

Finally, equation 4 can be used:

Hence, the approximate mass of the Milky Way Galaxy within the Sun's orbit is .

Appendix:

The value of a light year in meters can be determined by means of equation 5: